I am not sure whether above argument is completely correct. Even if it was correct, my problem to proceed is: after we glue some collection of maps, the domain $U$ would become not so "regular". I am not sure the intersection of $U$ and an open disk would still be simple connected.

$\begingroup$Probably I am missing something, but why aren't the columns of $T(\lambda)$ a basis for its image?$\endgroup$
– Fan ZhengJul 10 '18 at 4:24

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$\begingroup$@FanZheng: An obvious mistake. The rank I have in mind is something smaller than $m$. Thanks for pointing out.$\endgroup$
– user1101010Jul 10 '18 at 4:40

$\begingroup$Then it looks like the question "is every rank $l$ complex vector bundle over $\mathbb C$ trivial?", which is true because $\mathbb C$ is contractible.$\endgroup$
– Fan ZhengJul 10 '18 at 20:57

$\begingroup$I am aware of the argument to assert the existence. But more interested in the explicit construction by gluing together locally defined maps. Is this doable?$\endgroup$
– user1101010Jul 10 '18 at 21:47

$\begingroup$I guess it is precisely because of the combinatorial complications arising from the "gluing local data" argument that full-fledged theories of vector bundles, characteristic classes and sheaf cohomology are born.$\endgroup$
– Fan ZhengJul 11 '18 at 4:59

1 Answer
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I am not sure whether this gluing method will work, but as I mentioned in the comment, this problem is equivalent to "whether every rank $m$ (typo in the comment, sorry) complex vector bundle over $\mathbb C$ is trivial". To see this, first construct the trivial bundle of rank $n$ over $\mathbb C$, that is, $\mathbb C\times\mathbb C^n$. For every $x\in\mathbb C$, we have an $m$ dimensional subspace $E_x$ spanned by the columns of the matrix $T(x)$. By the local condition, there is a map $\phi_x:U(x)\times\mathbb C^m\to\cup_{y\in U(x)} E_y$, $(y,v)\mapsto h_x(y)v$. The transition maps $\phi_y^{-1}\circ\phi_x: (U(x)\cap U(y))\times\mathbb C^m\to(U(x)\cap U(y))\times\mathbb C^m$ are continuous, as the second paragraph of the question shows. Therefore $\phi_x^{-1}$ are local trivializations of $E:=\cup_{x\in\mathbb C} E_x$, which is then a complex vector bundle of rank $m$ over $\mathbb C$. The desired global continuous function $\phi$ similarly gives rise to a continuous function $\mathbb C\times\mathbb C^m\to E$ mapping $\{x\}\times\mathbb C^m$ to $E_x$, and hence establishes an isomorphism between $E$ and the trivial bundle $\mathbb C\times\mathbb C^m$.

Now the question reduces to "whether every rank $m$ complex vector bundle over $\mathbb C$ is trivial". The answer is Yes and the reasoning is the following. Since $\mathbb C$ is contractible, the identity map $i$ on $\mathbb C$ is homotopic to the map $j:\mathbb C\to\{0\}$. By Theorem 1.6 of http://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf, $E=i^*E$ is isomorphic to $j^*E=\mathbb C\times E_0$, which is trivial because $E_0$ is isomorphic to $\mathbb C^m$.

PS The OP asks the question in the continuous category (on which this answer is based) but in the comments the OP mentions the smooth category. It's an exercise left to the reader (and the OP) to carry the above argument over to the smooth category.

PPS The cited book (by Allen Hatcher, who also happens to be a MO user) is a wonderful introduction to vector bundles (and beyond).